L(s) = 1 | + (0.258 + 0.965i)2-s + (1.53 − 0.796i)3-s + (−0.866 + 0.499i)4-s + (0.136 + 2.23i)5-s + (1.16 + 1.27i)6-s + (2.35 − 1.20i)7-s + (−0.707 − 0.707i)8-s + (1.73 − 2.45i)9-s + (−2.12 + 0.709i)10-s + (−3.01 + 1.73i)11-s + (−0.933 + 1.45i)12-s + (−0.354 + 0.354i)13-s + (1.77 + 1.96i)14-s + (1.98 + 3.32i)15-s + (0.500 − 0.866i)16-s + (−3.12 − 0.838i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (0.887 − 0.459i)3-s + (−0.433 + 0.249i)4-s + (0.0611 + 0.998i)5-s + (0.476 + 0.522i)6-s + (0.889 − 0.456i)7-s + (−0.249 − 0.249i)8-s + (0.577 − 0.816i)9-s + (−0.670 + 0.224i)10-s + (−0.908 + 0.524i)11-s + (−0.269 + 0.421i)12-s + (−0.0982 + 0.0982i)13-s + (0.474 + 0.523i)14-s + (0.513 + 0.858i)15-s + (0.125 − 0.216i)16-s + (−0.758 − 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54151 + 0.693109i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54151 + 0.693109i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (-1.53 + 0.796i)T \) |
| 5 | \( 1 + (-0.136 - 2.23i)T \) |
| 7 | \( 1 + (-2.35 + 1.20i)T \) |
good | 11 | \( 1 + (3.01 - 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.354 - 0.354i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.12 + 0.838i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.42 - 1.39i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.64 + 1.51i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 31 | \( 1 + (4.44 + 7.69i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.09 - 2.16i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 7.96iT - 41T^{2} \) |
| 43 | \( 1 + (6.03 - 6.03i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.617 - 2.30i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.55 + 5.79i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (2.18 + 3.78i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.79 + 6.57i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.64 - 13.6i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 3.96iT - 71T^{2} \) |
| 73 | \( 1 + (-3.68 - 0.988i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.48 - 4.32i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.4 - 10.4i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.59 - 6.23i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.6 - 11.6i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.90886497201501633669934206088, −11.55604942612676066464689374806, −10.48196539442343166513024861087, −9.387007772884094658943338945446, −8.163240609602049448356189725122, −7.40482898601925816084509820082, −6.75272730054283949462377574083, −5.13947286879923261317874217134, −3.72574197089088489131344083334, −2.26920789853026370567478421808,
1.81155972671823520741813135596, 3.25077983589614827591850155066, 4.76712902690899456221339040858, 5.31566005833493545106469421807, 7.58348952998528700916300350937, 8.712195007701393487005134843329, 9.017774191910735502887887431986, 10.37068533036564522394327047898, 11.18924073881005750335150595305, 12.33498802565273358709503975550